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\centerline{\bf{Algebra Preliminary Exam}}

\centerline{\bf{January 20, 1995}}

\bigskip

\item{1.} Prove or disprove the following assertion: \newline
Every real symmetric matrix has a unique real symmetric cube root.

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\item{2.} Let ${\bold{Z}}[x]$ be the ring of polynomials in one variable with coefficients in the ring  ${\bold{Z}}$  of integers.  Let $I$ be the ideal of all polynomials  $f(x)$ in ${\bold{Z}}[x]$ such that  $f(0) = 0$, and let  $J$ be the ideal of all polynomials $f(x)$ in ${\bold{Z}}[x]$ such that $f(0)$ is an even integer.

Show that:

\item\item{a)} $I$ is a prime ideal.

\item\item{b)} $J$ is a maximal ideal.

\item\item{c)} $I$ is a principal ideal.

\item\item{d)} $J$ is {\bf{not}} a principal ideal.

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\item{3.} Let $E$ be the splitting field over the field $F$ of the polynomial $t^{15} - 1$.  Determine the extension degree $[E : F]$ when $F$ is:

(a) the field ${\bold R}$ of real numbers.

(b) the field ${\bold{F}}_2$ of integers mod 2.

(c) the field ${\bold{F}}_{31}$ of integers mod 31.

(d) the field ${\bold Q}$ of rational numbers.  {\bf Hint}:  The splitting fields of $t^5 - 1$ and of $t^3 - 1$ 

\hskip14pt are subfields.  Show that the intersection of these two subfields is ${\bold Q}$ .

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\item{4.} Let $E$ be a (finite) Galois extension field of  $F$ with Galois group $G$; let $K$ be an intermediate field and $H$ the subgroup of $G$ that fixes $K$.  Show that the subgroup of $G$ consisting of all $\sigma$ in $G$ for which  $\sigma(K) = K$ is the normalizer of $H$ in $G$.

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\item{5.}  Prove directly, without quoting the structure theorem for finitely generated modules, that in a principal ideal domain the matrices $\bigg[ \matrix
a & 0 \\
0 & b)
\endmatrix
\bigg]$ and $\bigg[ \matrix
gcd(a,b) & 0 \\
0 & lcm(a,b)
\endmatrix
\bigg]$ are row-and-column equivalent.

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\item{6.} Suppose that for a certain integer $n > 1$, every group of order $n$ is cyclic. Prove that $n$ is relatively prime to $\phi(n)$.

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\item{7.} Let $\ell$ denote ``length'' (in the sense of the Jordan-Holder theorem ).  Complete the following statement concerning the nontrivial abelian groups $A$ and $B$ ,and then prove the assertion:
$$\ell(A) = \ell(B) \text{if and only if} \ \dots \ .$$

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\item{8.} Let $R$ be a commutative ring. Let $M$ be a $R$-module. Consider the functors  $- \otimes  M$, Hom$(M,-)$ , Hom$(-,M)$  on the category of $R$-modules.

\item\item{(a)}  List the exactness properties of each of these three functors in general and prove 

\hskip14pt in detail what you have said about one of them.

\item\item{(b)}  What more can you say if $M$ is $R$-free?


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